As the swim coach at Springer, Daniel selects which athletes will participate in the state-wide swim relay. The relay team swims $\frac{6}{7}$ of a mile in total, with each team member responsible for swimming $\frac{3}{14}$ of a mile. The team must complete the swim in $\frac{3}{5}$ of an hour. How many swimmers does Daniel need on the relay team?
Solution: To find out how many swimmers Daniel needs on the team, divide the total distance ( $\frac{6}{7}$ of a mile) by the distance each team member will swim ( $\frac{3}{14}$ of a mile). $ \dfrac{{\dfrac{6}{7} \text{ mile}}} {{\dfrac{3}{14} \text{ mile per swimmer}}} = {\text{ number of swimmers}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{3}{14} \text{ mile per swimmer}}$ is ${\dfrac{14}{3} \text{ swimmers per mile}}$ $ {\dfrac{6}{7}\text{ mile}} \times {\dfrac{14}{3} \text{ swimmers per mile}} = {\text{ number of swimmers}} $ $ \dfrac{{6} \cdot {14}} {{7} \cdot {3}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $6$ in the numerator and the $3$ in the denominator by $3$ $ \dfrac{{\cancel{6}^{2}} \cdot {14}} {{7} \cdot {\cancel{3}^{1}}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $14$ in the numerator and the $7$ in the denominator by $7$ $ \dfrac{{2} \cdot {\cancel{14}^{2}}} {{\cancel{7}^{1}} \cdot {1}} = {\text{ number of swimmers}} $ Simplify: $ \dfrac{{2} \cdot {2}} {{1} \cdot {1}} = {4} $ Daniel needs 4 swimmers on his team.